Question

Given that \[\int {\dfrac{{dx}}{{\sqrt {2ax - {x^2}} }} = {a^n}{{\sin }^{ - 1}}\left[ {\dfrac{{x - a}}{a}} \right]} \] where a = constant. Using dimensional analysis, the value of n is:
A) $1$
B) $0$
C) $ - 1$
D) None of the above

Answer 1

Hint: Here in this question, we have to use a technique for analysis that, in the absence of sufficient data to establish accurate equations, expresses physical quantities in terms of their basic dimensions. As per given in the question we have to justify the value of n by using the dimensional analysis.




Complete answer:


We know that,
In order to determine the value using dimensional analysis, we must: Only if two physical quantities have the same dimensions can they be compared.
The multiplication of the dimensions of the two quantities yields the dimensions of the multiplication of two quantities.
As per the conclusion we get that, as given in the L.H.S. of the question we get a integral equation after solving that we only get a number. So, as from this the dimension of L.H.S. is $\left[ {{M^0}{L^0}{T^0}} \right]$
And Similarly in R.H.S we get that, there are also constant value so their dimensions is null as from this the dimensional formula of R.H.S is \[{a^n}\] .
Hence, by comparison both side we get that,
$\left[ {{M^0}{L^0}{T^0}} \right] = \left[ {{a^n}} \right]$
Here in the above equation, by comparison we get that, n belongs to the power of the dimensional formula of the L.H.S. equation. As form this the value of n is,
‘n’ must equal zero in order for the term to be dimensionless on both sides.
$n = 0$
Therefore, the correct answer is $n = 0$ .

Hence, the correct answer is Option B.





Note:We have to know that, Conversion between two sets of units is done via dimensional analysis, commonly known as the factor label method or unit analysis. This technique uses relationships or conversion factors between various sets of units to do simple conversions (e.g., feet to inches) and complex conversions (e.g., g/cm3 to kg/gallon).